a generalised airy distribution function for the accumulated...

J Stat Phys (2016) 162:1587–1607DOI 10.1007/s10955-016-1467-2

A Generalised Airy Distribution Function for theAccumulated Area Swept by N Vicious Brownian Paths

Isaac Pérez Castillo1 · Denis Boyer1

Received: 13 August 2015 / Accepted: 28 January 2016 / Published online: 15 February 2016© Springer Science+Business Media New York 2016

Abstract In this work exact expressions for the distribution function of the accumulated areaswept by excursions and meanders of N vicious Brownian particles up to time T are derived.The results are expressed in terms of a generalised Airy distribution function, containing theVandermonde determinant of the Airy roots. By mapping the problem to an Random MatrixTheory ensemble we are able to performMonte Carlo simulations finding perfect agreementwith the theoretical results.

Keywords Random walkers · Vicious walkers · Random matrices · Airy distributionfunction

1 Introduction

Since the seminal work of de Gennes on simple models of fibrous structures and its pop-ularisation by Fisher, the study of vicious walkers (namely, walkers that do not intersect)have attracted attention during the last two decades due to the wide range of appli-cations in various branches of science and their connections to random matrix theory[1,2,4,5,7,9,12,14,16,17,27–29,31–33,37]. Similarly, the statistical properties of the areaswept by a Brownian excursion is a problem that was originally studied by mathematicians.The Laplace transform of the distribution of the area (known as the Airy distribution func-tion) was first computed by Darling and Louchard [8,22]. The derivation of its momentstogether with the actual distribution were obtained by Takács [34–36]. The Airy distributionfunction has appeared in a number of problem from different areas from graph theory andcomputer science [11,15,24] as well as in physical systems modelling one dimensional fluc-tuating interfaces [23,25], applications in laser cooling [3,19] and fluctuations of sizes ofring polymers [26].

B Isaac Pérez [email protected]

1 Departamento de Sistemas Complejos, Instituto de Física, UNAM, P.O. Box 20-364, 01000 Mexico,D.F., Mexico

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1588 I. P. Castillo, D. Boyer

In this work, we focus on obtaining exact expressions for the accumulated area swept by NviciousBrownianmotions. This paper is organised as follows: in Sect. 2,we define the processof vicious walkers and the stochastic quantities we are interested in. In particular, we showin Sect. 3 that the Laplace transform of the PDF of the accumulated area can be expressedas a ratio of two N-particle propagators, which can be constructed as Slater determinants ofsingle-particle states. This is a straightforward consequence of using Quantum Mechanicsformalism (QMf) or, alternatively, of the Karlin-McGregor formula. The resulting expressionis fairly general. In order to extract some properties we consider two types of processes:excursions and meanders. Moreover we consider two types of boundary conditions at x = 0:absorbing boundary conditions (Sect. 4) and reflecting boundary conditions (Sect. 5). Theinverse Laplace transform is resolved in Sect. 6 and a formula for its negative moments isobtained in Sect. 7. Finally, in Sect. 8, using a mapping between processes of vicious walkersto ensembles of RandomMatrix Theory we are able to performMonte Carlo simulations andcompare with the exact formulas obtained in the previous sections.

2 Model Definitions

Consider the set of trajectories defined by N one-dimensional Brownian particles x(τ ) =(x1(τ ), . . . , xN (τ )) with τ ∈ [0, T ] such that 0 < x1(τ ) < · · · < xN (τ ) and let us denote asMN [x(τ )] its probability measure. Next, let us define as Ai the area swept by particle i :

Ai =∫ T

0xi (τ )dτ .

The set of areas A = (A1, . . . , AN ) is a random N -tuple whose joint PDF (jPDF) is givenby viz.

PN (A, T ) =∫ x(T )=x f

x(0)=xiDx(τ )MN [x(τ )]

N∏i=1

δ

(∫ T

0xi (τ )dτ − Ai

). (1)

Notice that the jPDF (1) contains the PDF of the area swept for each curve i = 1, . . . , N

P(i)N (Ai , T ) =

∫ ∞

0. . .

∫ ∞

0d A1 . . . d Ai−1d Ai+1 . . . d ANPN (A, T ) , (2)

and, in particular, the PDFs of the area for the bottom and top curves denoted as TN (A, T ) ≡P(N )N (A, T ) and BN (A, T ) ≡ P(1)

N (A, T ), respectively. It also contains the PDF of the totalarea, defined as

SN (A, T ) =∫ ∞

0. . .

∫ ∞

0d A1 . . . d ANPN (A, T )δ

(A −

N∑i=1

Ai

). (3)

Exact formulas of the expected values of the areas for the top and bottom curve can beenfound in [37], but as far as we are aware of, the exact formulas exist for their distributions.In this work we present exact formulas for the PDF SN (A, T ) for the accumulated area andcompare the expressions with Monte Carlo simulations. Notice that the expression for thePDF of accumulated area shares some similarities to the terrace-legde-kink model whereslope variations are enforced through a volume constraint [10]. A thorough analysis basedon Monte Carlo simulations on the marginals P(i)

N (A, T ) for i = 1, . . . , N will be discussedsomewhere else.

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A Generalised Airy Distribution Function for the Accumulated... 1589

3 Path Integral Approach

Combining (1) and (3) yields

SN (A, T ) =∫ x(T )=x f

x(0)=xiDx(τ )MN [x(τ )]δ

(A −

N∑i=1

∫ T

0xi (τ )dτ

),

with

M[x(τ )] = 1

Zexp

[−1

2

N∑i=1

∫ T

0dτ

(dxi (τ )

dτ

)2]CN [x(τ )] ,

where Z is the normalisation factor and CN [x(τ )] is an indicator function imposing theconstraint of vicious particles 0 < x1(τ ) < · · · < xN (τ ) on the positive part of the real line.Further, the Laplace transform of SN (A, T ),

SN (λ, T ) =⟨e−λA

⟩SN

≡∫ ∞

0d A SN (A, T )e−λA

has the following representation using path integral approach:

SN (λ, T ) = ∫ x(T )=x f

x(0)=xiDx(τ )MN [x(τ )] exp

[−λ

∑Ni=1

∫ T0 xi (τ )dτ

]. (4)

In QMf, the expression (4) can be expressed as the ratio of two propagators

SN (λ, T ) = G(1)N (x f ,T |xi ,0)

G(0)N (x f ,T |xi ,0)

, (5)

with G(a)N (x, T |x0, 0) =

⟨x|e−H (a)

N T |x0⟩and Hamiltonian operators

H (a)N =

N∑j=1

h(a)j = −1

2

N∑j=1

∂2x j +N∑j=1

V (a)(x j ) , a = 0, 1 (6)

and V (0)(x) a confining potential in R+, while V (1)(x) = λx for x > 0 and infinity for

x ≤ 0. Besides, Brownian particles are not allowed to cross which in QMf corresponds tohaving fermions. Considering the latter and that the potentials are one-particle type opera-tors, the eigenfunctions of H (a)

N are given by the Slater determinant of the eigenfunctionscorresponding to the one-particle problem h(a), that is

G(a)N ( y, T |x, 0) = ∑

n�

(a)n ( y)�

(a)

n (x)e−E (a)n T , (7)

with

�(a)n (x) = 1√

N ! det1≤i, j≤N

φ(a)ni (x j ) , E (a)

n =N∑i=1

e(a)ni , h(a)

∣∣∣φ(a)n

⟩= e(a)

n

∣∣∣φ(a)n

⟩.

Once the corresponding one-particle problems of the numerator and denominator have beensolved, we have solved the problem via the formula (5). However, the corresponding expres-sion is still too general to extract general mathematical properties. Thus, we will particularise(5) to the cases of excursions and meanders. For excursions we mean the set of processes bywhich the N vicious Brownian paths start and finish at the same point, that is xi = x f = 0.

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This is achieved by taking xi = x f = ε with ε = (ε1, · · · , εN ), and taking the limit ε to 0,viz.

S(r)N (λ, T ) = lim

ε→0

G(1)N (ε,T |ε,0)

G(0)N (ε,T |ε,0) . (8)

For meanders we mean the set of processes where the paths start at xi = 0 and are allowedto finish at any final position x f at time T . As such we need to integrate over all possiblefinal positions and, therefore, the Laplace transform of the corresponding PDF reads

S(m)N (λ, T ) = lim

ε→0

∫W N

dN x f G(1)N (x f , T |ε, 0)∫

W NdN x f G

(0)N (x f , T |ε, 0)

, (9)

where W N stands for the Weyl chamber W N = {x ∈ R

N |0 ≤ x1 ≤ · · · ≤ xN}and dN x =

dx1 · · · dxN .

4 Absorbing Boundary Conditions

We note in the two cases above that we will need the one-particle problem for a = 1 (the one-particle problem for a = 0 is treated in the appendices). For absorbing boundary conditions(i.e. the corresponding eigenfunctions of the one-particle problem obey φ(x = 0) = 0) ithas the well-known solution

φ(1)n (x) =

√(2λ)1/3

|Ai′(−αn)|Ai[(2λ)1/3x − αn] , e(1)n = αnλ

2/32−1/3 , n = 1, 2, . . .

where Ai(x) is the Airy function with zeros z = −αn with n ∈ {1, 2, . . .}. Thus the eigen-function for the N -particle problem is:

�(1)n (x) = 1√

N ! det1≤i, j≤N

φ(1)ni (x j ) = (2λ)N/6√

N !∏Ni=1 |Ai′(−αni )|

det1≤i, j≤N

[ϕni (z j )

], (10)

where we have defined ϕn(z) = Ai(z − αn) with z = (2λ)1/3x .

4.1 Excursions

To obtain an exact expression for S(r)N (λ, T ) given by (8) we need to perform the Taylor

expansion of the Slater determinant (10) containing the Airy eigenfunctions. To do so wemake use of the following formula [21]:

Ai(t)Bi(z + t) − Bi(t)Ai(z + t) = 1

π

∞∑�=0

z�

�! Q�(t) ,

with Qn(z) the Abramochkin polynomials [21]. In our case, we take that t = −αn whichyields

Ai(z − αn) = Ai′(−αn)

∞∑�=0

z�

�! Q�(−αn) ,

where we have used the identity πBi(−αn)Ai′(−αn) = −1. Combining this together withthe standard formula of the Taylor series of a Slater determinant (see Appendix 1), we arriveat

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det1≤i, j≤N

ϕni (z j ) =[

N∏j=1

Ai′(−αn j )

] ∑�∈UN

(2λ)13∑N

j=1 � j

�1!...�N ! det1≤i, j≤N

[x� ji ] det

1≤i, j≤N[Q�i (−αn j )] ,

(11)

where � ∈ UN stands for the sum over all ordered N -tuple indices � = (�1, . . . , �N ), that iswith �1 < �2 < . . . < �N .

Everything boils down to be able to find an expression for the determinant det1≤i, j≤N

[Q�i

(−αn j )] of the Abramochkin polynomials evaluated at the Airy roots. It turns out that it ispossible to obtain the lowest contribution in ε if we recall that the odd-numbered polynomialsare such that Q2n+1(z) = zn + lower order terms, while even-numbered polynomials canbe expressed in terms of the previous odd-numbered ones. This automatically implies thatthe first non-zero lowest-order contribution in ε in the sum in Eq. (11) is given by taking� j = 2( j − 1) + 1 for j = 1, . . . , N . This eventually yields:

det1≤i, j≤N

ϕni (z j ) ∼ (2λ)N23

[N∏j=1

Ai′(−αn j )

][N∏i=1

xi

] N (x21 , . . . x

2N ) N (αn1 , . . . αnN ) ,

(12)

where we have defined γN = (−1)N (N−1)

2 /[1! · · · (2N + 1)!], and N (a1, . . . , aN ) is theVandermonde determinant:

N (a1, . . . , aN ) ≡

∣∣∣∣∣∣∣∣∣

1 · · · 1a1 · · · aN...

. . ....

aN−11 · · · aN−1

N

∣∣∣∣∣∣∣∣∣=

∏1≤i< j≤N

(ai − a j ) .

To obtain the expression (12) we have used that

det1≤i, j≤N

[x2( j−1)+1i ] =

[N∏i=1

xi

] N (x21 , . . . x

2N ),

det1≤i, j≤N

[Q�i (−αn j )] = (−1)N (N−1)

2 N (αn1 , . . . αnN ) .

The final expression of the Taylor expansion for the N -particle eigenfunction is:

�(1)n (ε) ∼ βN (2λ)

N (2N+1)6

[N∏j=1

Ai′(−αn j )

|Ai′(−αn j )|

][N∏i=1

εi

] N (ε21 , . . . ε

2N ) N (αn1 , . . . αnN ) ,

(13)

with βN = γN/√N !. A similar derivation can be done for the denominator (see Appendix 2).

Using the results (13), (30), and (39) in the corresponding expression (7) of the propagator, andafter rearranging terms, we eventually obtain the Laplace transform of the PDF S(r)

N (A, T ),viz.

S(r)N (λ, T ) =

(λT 3/2

)E(r)N

A(r)N

∑n1,...,nN 2

N (αn1 , . . . αnN )e−(λT 3/2)2/32−1/3∑Ni=1 αni ,

A(r)N = 2

12 E(r)

N

πN

N−1∏j=0

�(2 + j)�(3/2 + j) , E (r)N = 1

3N (2N + 1) . (14)

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There is a the striking resemblance of the result (14) with that found for the distributionof the maximal height of N-non intersecting Brownian excursions obtained in [33]. Thissimilarity is essentially due to Pauli’s exclusion principle as the vicious walkers cannotcross, which results in appearance of the Vandermonde determinant in both expressions.However, whenever a probability of crossing between particles is allowed, as for instance in[6], a generalized Vandermonde determinant appears instead.

4.2 Meanders

For the case (9) we report the final result (the derivation can be found in Appendix 3)

S(m)N (λ, T ) =

(λT 3/2

)E (m)N

A(m)N

∑n1,...,nN

B(A)N (αn1 , . . . , αnN )

N (αn1 , . . . αnN )e−(λT 3/2)2/32−1/3∑Ni=1 αni , (15)

where

B(A)N (αn1 , . . . , αnN ) = 1∏N

i=1 Ai′(−αni )

∫W N

dN x det1≤i, j≤N

[Ai(x j − αni )

], E (m)

N = N 2

3.

In this case we were unable to find a simple expression for the normalisation constant A(m)N .

5 Reflecting Boundary Conditions

So farwehave preoccupied ourselves by considering absorbing boundary conditions at x = 0.We have also considered how the preceding derivations change when we take reflectingboundary conditions instead. Apart from the mathematical curiosity, the resulting process isinteresting as it is related to the work [17] for processes X (−1/2,0)(t) which show a transitionfrom type D to type D’ matrix ensembles (see [17] for details).The first thing is to notice how the one-particle wave-function changes in this case. If wedenote as z = −βn with n ∈ 1, 2, . . . the zeros of the derivative of the Airy function Ai′(x),then

φ(1)n (x) =

√(2λ)1/3√

βn |Ai(−βn)|Ai((2λ)1/3x − βn) , e(1)n = βnλ

2/32−1/3

Thus, in this case the N -particle wavefunction takes the following form:

�(1)n (x) = (2λ)N/6

√N !∏N

i=1

√βni |Ai(−βni )|

det1≤i, j≤N

[ψni (z j )],

ψn(z) = Ai(z − βn) , z = (2λ)1/3x .

5.1 Excursions

We are left with deriving a Taylor expansion of the Slater determinant, as before. In this case,the other Abramochkin polynomials [21] become useful:

Bi′(z)Ai(z + t) − Ai′(z)Bi(z + t) = 1

π

∞∑n=0

tn

n! Pn(z) .

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From this expression we choose t = −βn to write

Ai(z − βn) = Ai(−βn)

∞∑�=0

t�

�! P�(−β�) ,

where we have used the property πBi′(−βn)Ai(−βn) = 1. The expansion of the Slaterdeterminant yields

det1≤i, j≤N

ψni (z j ) =[

N∏j=1

Ai(−βn j )

]∑�∈UN

(2λ)13∑N

j=1 � j

�1!···�N ! det1≤i, j≤N

[x� ji ] det

1≤i, j≤N[P�i (−βn j )] ,

Again here, the lowest contribution in ε is given by the even-labelled polynomials P2n(z) =zn + · · · for n = 0, 1, 2 . . . and with P0(z) = 1. Thus we must take � j = 2( j − 1) forj = 1, . . . , N yielding the following result for the Slater determinant

�(1)n (ε) = δN (2λ)

N (2N−1)6

⎡⎣ N∏

j=1

Ai(−βn j )√βn j |Ai(−βn j )|

⎤⎦ N (x21 , . . . , x

2N ) N (βn1 , . . . , βnN )

with δN = (−1)N (N−1)

2 /[0!2! · · · 2(N − 1)!√N !]. A similar analysis can be done to thepropagator in the denominator of Eq. (5) (see Appendix 2) eventually obtaining

S(r)N (λ, T ) =

(λT 3/2

)E(r)N

A(r)N

∑n1,...,nN

2N (βn1 ,...,βnN )

βn1 ···βnN e−(λT 3/2)2/32−1/3∑Ni=1 βni

A(r)N = 2

12 E(r)

N

πN

N−1∏j=0

�( 12 + j

)� ( j + 2) , E (r)

N = 13N (2N − 1) ,

5.2 Meanders

Similarly, in the case of meanders (see Appendix 3), we obtain

S(m)N (λ, T )=

(λT 3/2

)E (m,R)N

A(m)N

∑n1,...,nN

B(R)N (βn1 , . . . , βnN )

× N (βn1 , . . . βnN )

βn1 · · · βnNe−(λT 3/2)2/32−1/3∑N

i=1 βni , (16)

where

B(R)N (βn1 , . . . , βnN ) = 1∏N

i=1 Ai(−βni )

∫W N


[Ai(x j − βni )

],

E (m)N = 1

3N (N − 1) .

and with no simple expression for the normalisation constant A(m)N .

6 Inverse Laplace transform of SN(λ, T)

First of all, we start by noticing that SN (λ, T ) = QN (s) with s = λT 3/2. This, as pointedout already in [25], implies a scaling law of the form SN (A, T ) = T−3/2QN (AT−3/2) so

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Table 1 Exponents E(r)N and E(m)

N as a function of N for both absorbing and reflecting boundary conditions

No. ofParticles

Absorbing boundary conditions Reflecting boundary conditions

N E(r)N n k E(m)

N n k E(r)N n k E(m)

N n k

1 1 1 0 13 – – 1

3 – – 0 0 0

2 103 2 2 4

3 0 2 2 2 0 23 0 1

3 7 7 0 3 3 0 5 5 0 2 2 0

4 12 12 0 163 4 2 28

3 8 2 4 4 0

5 553 17 2 25

3 7 2 15 15 0 203 6 1

6 26 26 0 12 12 0 22 22 0 10 10 0

7 35 35 0 493 15 2 91

3 29 2 14 14 0

8 1363 44 2 64

3 20 2 40 40 0 563 18 1

9 57 57 0 27 27 0 51 51 0 24 24 0

10 70 70 0 1003 32 2 190

3 62 2 30 30 0

In the columns n and k, we show the number of derivatives involved for the function F(x, γ ). Recall that the

expressions for the exponents are E(r)N = N (2N+1)/3 and E(m)

N = N2/3 forABCs and E(r)N = N (2N−1)/3

and E(m)N = N (N − 1)/3 for RBCs

that QN (s) = L[QN (x)] with x = AT−3/2. This applies to both cases of excursions andmeanders. Notice that, even though this scaling appears mathematically, it can be easilyderived by simple dimensional analysis. Indeed, as the dimensions of the diffusion constantare [D] = L2T−1 and the dimensions of area in our problem are [A] = LT (Nb. here our Trefers to the time dimension, not to be confused with our final time T ) then we have that

SN (A, T ) = 1√DT 3

QN

(A√DT 3

),

as we have found1. Secondly, by looking at the exponents E (r)N and E (m)

N (see either at table 1or the expressions for the exponents in Eqs. (14) and (16), it is clear that we must performthe inverse Laplace transform L−1[sδes

2/3 ] with δ either an integer or one-third of an integer.To this end, we start from the result∫ ∞

0F(x)e−sxdx = e−s2/3 , F(x) = 21/3 J (21/3x−2/3)x−5/3 ,

with

J (x) = 22/3x

33/2√

πU (1/6, 4/3, 2x3/27)e− 2x3

27 ,

and with U (a, b, x) the confluent hypergeometric function. After doing the rescalling ofs2/3 → 2−1/3γ s2/3, and a change of variables, we obtain the following formula

sn+2k/3e−γ 2−1/3s2/3 = (−1)k2k/3∫ ∞

0

[∂k+n F(x, γ )

∂γ k∂xn

]esxdx,

F(x, γ ) ≡ γ J (γ x−2/3)x−5/3 ,

1 For simplicity we have set D = 12 . This can be thought as equating dimensions of length squared with time,

which implies that the dimensions of area are [A] = T 3/2

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or alternatively

L−1[sn+2k/3e−γ 2−1/3s2/3 ] = (−1)k2k/3∂k+n

∂γ k∂xnF(x, γ ) . (17)

Here we have used the property

L[F (n)(x)] = sn F(s) −n−1∑k=0

sk F (n−1−k)(0+) ,

with notation F (n)(x) = dnF(x)/dxn , and the fact that any derivative of F(x, γ )with respectto x at x → 0+ is zero.To get an idea of the order of the derivatives involved in the function F(x, γ ), one can seehow the exponents E (r)

N and E (m)N vary with N and choose a value of the pair (n, k) which

gives such an exponent. This choice is not necessarily unique and we show one possiblechoice in Table 1. The choice made is such that we only need derivatives with respect to γ

up to second order.The only case for which we are unable to use this prescription directly is for the case

N = 1 for meanders, as this will imply the exponent E (m)1 = 1/3. A way around this is to

consider in this case the pair (0, 2) which will give the derivative of the PDF, instead.This being settled, we proceed to find a simple expression for any order derivative of the

function F(x, γ ). Starting from

F(x, γ ) =√3

x√

πu2/3(x, γ )e−u(x,γ )U

(1

6,4

3, u(x, γ )

), u(x, γ ) = 2γ 3

27x2,

and using properties of the hypergeometric confluent function we arrive at the followingresult (see Appendix 5)

∂k+n F(x, γ )

∂γ k∂xn=

√3

xn+1√

πγ ku2/3(x, γ )e−u(x,γ )

n∑�=0

k∑s=0

C (n)� D(k)

s (�)U

(1

6− � − s,

4

3, u(x, γ )

), (18)

where the set of coefficients {C (n)� } and {D(k)

s (�)} are given by

C (n)� = n!

2n−2�(n − �)!(2� − n)! , � = 0, . . . , n ,

D(0)0 (�) = 1 , D(1)

0 (�) = −3

2(1 + 2�) , D(1)

1 (�) = −3

D(2)0 (�) = 3

4(1 + 2�)(5 + 6�) , D(2)

1 (�) = 3(7 + 6�) , D(2)2 (�) = 9 .

Notice that for the set of coefficients {D(k)s (�)} we have already taken into account the fact

that we only need derivatives with respect to γ up to second order.With the help of Eqs. (17) and (18), we can now perform the inverse Laplace transform

of Q(r)N (s) and Q(m)

N (s), obtaining the following generalised Airy distributions for absorbingboundary conditions:

Q(r)N (x) =

∑n1,...,nN

2N (αn1 , . . . αnN )I(n,k)

N ,(r)

(x,

N∑i=1

αni

), (19)

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Q(m)N (x) =

∑n1,...,nN

B(A)N (αn1 , . . . αnN ) N (αn1 , . . . αnN )I(n,k)

N ,(m)

(x,

N∑i=1

αni

). (20)

For reflecting boundary conditions we have instead

Q(r)N (x) =

∑n1,...,nN

2N (βn1 , . . . βnN )

βn1 . . . βnNI(n,k)N ,(r)

(x,

N∑i=1

βni

), (21)

Q(m)N (x) =

∑n1,...,nN

B(R)N (βn1 , . . . βnN )

N (βn1 , . . . βnN )

βn1 . . . βnNI(n,k)N ,(m)

(x,

N∑i=1

βni

). (22)

In both cases we have defined

I(n,k)N ,(s)(x, γ ) = (−1)k2k/3

A(s)N

√3

xn+1√


n∑�=0

k∑s=0

C (n)� D(k)

s (�)U

(1

6− � − s,

4

3, u(x, γ )

).

with s ∈ {(excu)r(sions),m(eanders)}. In the expression of I(n,k)N ,(s)(x, γ ) the pair of indices

(n, k)must be chosen according to the values related to the exponents E (r)N and E (m)

N appearing

in table 1. Notice, in particular, that for N = 1 and (n, k) = (1, 0) we have that A(r)1 =

1/√2π , N = 1, C (1)

0 = 0, C (1)1 = 2. Thus Q(r)

1 (x) we recover to the so-called Airydistribution, viz

Q(r)1 (x) = ∑∞

n=1 I(1,0)1,(r) (x, αn) , I(1,0)

1,(r) (x, γ ) = 2√6

x2

(2γ 3

27x2

)2/3e− 2γ 3

27x2 U(− 5

6 ,43 ,

2γ 3

27x2

).

7 Moments

In this section we discuss the derivation of the moments. We consider negative powered-moments. Let us start by fixing some notation. Let us denote Mn the n-th moment for aPDF F(x) with x ∈ R

+ An = ∫∞0 dxxn F(x). Suppose next that the Laplace transform of

a function F(x) is F(s) = sEe−as2/3 , with E and a two constants. Then one can show that(see Appendix 6)

A−ν = 3

2a− 3

2 (ν+E)�( 32 (ν + E)

)�(ν)

.

In our case let us denote the moments as M (a)N ,n = ∫∞

0 dxQ(a)N (x)xn with a ∈ {r,m}. Using

the previous results, we obtain the following formulas for the negative moments of the PDFfor excursions and meanders with ABCs

123


M (r)N ,−ν = 2

ν+E(r)N

2 −13�(32 (ν + E (r)

N ))

A(r)N �(ν)

∑n1,...,nN

2N (αn1 , . . . , αnN )

(N∑i=1

αni

)− 32 (ν+E (r)

N )

M (m)N ,−ν = 2

ν+E(m)N

2 −13�(32 (ν + E (m)

N ))

A(m)N �(ν)

∑n1,...,nN

N (αn1 , . . . , αnN )BN (αn1 , . . . , αnN )

(N∑i=1

αni

)− 32 (ν+E (m)

N )

Similar expressions can be found for RBCs.

8 Monte Carlo Simulations

To check the correctness of our analytical findings, we have performed Monte Carlo simula-tions exploiting the connection with Random Matrix Theory [17,20] to generate samples ofnon-colliding paths. Following the notation in [20] we first recall the definition of the Paulimatrices

σ1 =(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

),

while we denote the 2×2 identity matrix I2 as σ0. For meanders we define the following2N × 2N Hermitian matrices �C

T (t) and �DT (t) corresponding to absorbing and reflecting

boundary conditions

�CT (t) = ia(0)

T (t, O) ⊗ σ0 + s(1)(t) ⊗ σ1 + s(2)T (t, O) ⊗ σ2 + s(3)(t) ⊗ σ3 ,

�DT (t) = ia(0)

T (t, O) ⊗ σ0 + ia(1)T (t, O) ⊗ σ1 + ia(2)(t) ⊗ σ2 + s(3)(t) ⊗ σ3 ,

respectively.Here the notation is a bit involved but itmeans the following: (· · · )T (t, O) standsfor bridges starting at the origin and finishing at the origin at time T , while a(a) and s(a)

denote N×N antisymmetric and symmetricmatrices, respectively, whose elements are eitherbridges or standard Brownian motions. Being more precise, if we denote as b(t) a standardBrownian motion with b(0) = 0, and with B(t) a Brownian bridge with B(0) = B(T ) = 0then

s(a)i j (t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

b(a)i j (t)/

√2 i < j

b(a)i i (t) i = j

b(a)j i (t)/

√2 j < i

, a(a)i j (t) =

⎧⎪⎨⎪⎩b(a)i j (t)/

√2 i < j

0 i = j

−b(a)j i (t)/

√2 j < i

and

s(a)T,i j (t, O) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

B(a)i j (t)/

√2 i < j

B(a)i i (t) i = j

B(a)j i (t)/

√2 j < i

, a(a)T,i j (t, O) =

⎧⎪⎨⎪⎩

B(a)i j (t)/

√2 i < j

0 i = j

−B(a)j i (t)/

√2 j < i

.

Here the index a = 0, 1, 2, 3, simply states that we need to construct different matrices.Similarly, for the case of excursions we have instead the following two 2N × 2N Hermitian

123


xi( )

i=1

i=2

i=3

i=4

x1( )

Bottom Curve

i=1

x3( )

Inner Curvei=3

x4( )

Top Curvei=4

xi( )

i=1

i=2i=3

i=4

x1( )

Bottom Curve

i=1

x3( )

Inner Curve

i=3

x4( )Top Curve

i=4

Fig. 1 Two instances corresponding to an excursion (top panel) and a meander (bottom panel) of N Viciouspaths with reflective boundary conditions at x = 0, generated by the method explained in the text. We alsoshow the corresponding areas for the bottom, one of the inners, and the top curves

matrices for absorbing and reflecting boundary conditions

�CT (t) = ia(0)

T (t, O) ⊗ σ0 + s(1)T (t, O) ⊗ σ1 + s(2)

T (t, O) ⊗ σ2 + s(3)T (t, O) ⊗ σ3 ,

�DT (t) = ia(0)

T (t, O) ⊗ σ0 + ia(1)T (t, O) ⊗ σ1 + ia(2)

T (t, O) ⊗ σ2 + s(3)T (t, O) ⊗ σ3 ,

respectively.Bridges are easily generated from a standard Brownian motion. In its discrete version if

R(k) is a standard random walk with k ∈ {0, . . . , K } with R(k = 0) = 0 then a discreteBrownian Bridge is given by B(k) = R(k) − (k/K )R(k).

To estimate the different PDFs we have taken K = 103 and we have subsequently gener-ated N = 105 �(t) matrices for the four different process. For each instance s = 1, . . . ,Nthe area swept by each path is estimated as

A(s)i =

K∑k=0

λ(s)i (k) , i = 1, . . . , N . (23)

where λ(k) = (−λN (k), . . . , −λ1(k), λ1(k), . . . , λN (k)) are the eigenvalues of the matrix�. This area is then rescaled to the x-variable x (s)

i = A(s)i /K 3/2. Finally, the sample set

{x (s)i }Ns=1 for each area i = 1, . . . , N , is used to estimate the PDFs as:

P(i)N (x) = 1

N

N∑s=1

δ(x − x (s)

i

)i = 1, . . . , N . (24)

Here we focus on the PDF QN (x) of the accumulated area. An instance of the four processescan be found in Figs. 1 and 2.

Results of the PDFs estimated by Monte Carlo simulations and comparison with thetheoretical formulas for the PDF QN (x) of the accumulated area are reported in Fig. 3 forboth types of boundary conditions.s

Looking at these figures some points are in order: the comparison between the set offormulas (19) and (21) and Monte Carlo estimates could be done up to N = 10 and N = 9,

123


xi( )

i=1 i=2

i=3

i=4

x1( )

Bottom Curvei=1

x3( )

Inner Curvei=3

x4( )

Top Curvei=4

xi( )

i=1

i=2

i=3i=4

x1( )

Bottom Curve

i=1

x3( )

Inner Curve

i=3

x4( )Top Curve

i=4

Fig. 2 Two instances corresponding to an excursion (top panel) and a meander (bottom panel) of N Viciouspaths with absorbing boundary conditions at x = 0 and generated by the method explained in the text. Wealso show the corresponding areas for the bottom, one of the inners, and the top curves

2 4 6 14 15 16 17 18x

0.5

1.0

1.5

2.0

2.5

N(r)(x)

N=1

N=2N=3

N=4N=5

N=10

\ \ 0 52

5 152

10 20 22 24 26x

0.20.40.60.81.01.2

N(m)(x)

N=10N=5

N=4N=3

N=2

N=1

\ \

Fig. 3 Plots for Q(r,m)N (x) for N = 1, 2, 3, 4 and 10. The top plots corresponds to absorbing boundary

conditions, while the bottom plots are for reflecting boundary conditions. Thick solid lines the theoreticalresults given by (19). All PDFs have been estimated by generating 105 samples

respectively, as otherwise the numerical evaluation of these exact formulas becomes pro-hibitively long. For the formulas (20) and (22) we have the an additional numerical problemdue to the evaluation of the BN and, as a consequence, the comparison with Monte Carlosimulations is performed up to N = 2 in both cases.

9 Conclusions

In this work we have generalised the Airy distribution function of the area swept by oneBrownian particle performing an excursion to the case of the accumulated area swept by

123


N vicious Brownian particles. The exact formulas have been contrasted with Monte Carlosimulations showing perfect agreement.

There are several open problemswhich we are currently investigating. First of all, whereasobtaining exact expressions for negative moments of the distribution of the accumulated areaswept by N vicious walkers is rather simple, wewonder whether it is possible to obtain exactsexpressions for the positive moments, perhaps generalising the techniques used for the caseof one walker that can be found, for instance, in [18,34,35]. Secondly, we ponder whether itis possible to find exact formulas for the distribution of the area swept by either the bottomor the top paths. Within QMf this entails to being able to construct N -particle eigenfunctionsfor distinguishable fermions. As this seems to be a daunting task, at the very least it wouldbe interesting to study the properties of these distributions numerically. This can be donefairly efficiently, in particular for rate events, by combining the mapping to RMT we haveused here together with the Wang-Landau algorithm, a numerical technique that has beenused in a similar context to study the statistics of extreme eigenvalues [30]. This analysiscan obviously be extended to the cases in which the distribution of jumps of the Brownianmotion comes from either thick or thin tails. TheMonte Carlo simulations could also be usedto explore the scaling of the PDFs of the areas for large N . An initial numerical analysisseems to indicate that the expected area of the i th curve goes like ∼ i/

√N , which implies

that the expected accumulated area scales like ∼ N 3/2.

Acknowledgments The authorswarmly thankN.Kobayashi andM.Katori for email correspondence regard-ing the simulations. We also thank E. Barkai for pointing out some references.

Appendix 1: Taylor Expansion of Slater’s Determinant

In this secion we discuss the Taylor expansion of a Slater determinant. Starting form thedefinition of the determinant and doing a Taylor expansion we write:

det1≤i, j≤N

ϕni (z j ) =∑P∈SN

sign(P)

N∏i=1

ϕnP (i)(zi )

=∑P∈SN

sign(P)∑

�1,...,�N≥0

z�11 . . . z�NN�1! . . . �N !

N∏i=1

ϕ(�i )nP(i)

(0)

=∑

�1,...,�N≥0

z�11 . . . z�NN�1! . . . �N ! det

1≤i, j≤Nϕ(�i )n j

(0) . (25)

Due to the antisymmetric properties of the determinant, only those terms in the multiple sumwith different values of �’s are different from zero. With this in mind we rewrite the sum overthe � = (�1, . . . , �N ) as a sumover overUN of ordered indices (let’s say �1 < �2 < · · · < �N )and then we sum over its permutation, viz.

det1≤i, j≤N

ϕni (z j ) =∑

�1,...,�N≥0

z�11 . . . z�NN�1! . . . �N ! det

1≤i, j≤Nϕ(�i )n j

(0)

=∑

�∈UN

1

�1! . . . �N !∑P∈SN

z�P(1)1 . . . z

�P(N )

N det1≤i, j≤N

ϕ(�P(i))n j (0)

123


=∑

�∈UN

1

�1! . . . �N !∑P∈SN

sign(P)z�P(1)1 . . . z

�P(N )

N det1≤i, j≤N

ϕ(�i )n j

(0)

=∑

�∈UN

1

�1! . . . �N ! det1≤i, j≤N

[z� ji ] det

1≤i, j≤N[ϕ(�i )

n j(0)] , (26)

as we wanted to show.

Appendix 2: Excursion

For the normalisation factor we need to derive the propagator of N vicious walkers in thesemi-infinite line. This propagator is this case can be written as follows

G0( y, T |x, 0)= 1

T N/2

(2

π

)N∫ ∞

0. . .

∫ ∞

0dq1 . . . dqN�(0)

q

(y√T

)�(0)

q

(x√T

)e− 1

2 q2,

(27)

where �(0)q (x) = (1/

√N !) det

1≤i, j≤N[sin(qi x j )]. Using the Taylor expansion formula for the

Slater determinant we arrive at

�(0)q (x) = 1√

N !∑s∈UN

1

(2s1 + 1)! . . . (2sN + 1)! det1≤i, j≤N

[x2s j+1i ] det

1≤i, j≤N[q2s j+1

i (−1)s j ] ,

(28)

with s = (s1, . . . , sN ). Noticing that the lowest contribution is obtained by of si = i − 1, weobtain

�(0)q

(x√T

)� βN T− N2

2

[N∏i=1

xi

] N

(x21 , . . . , x

2N

) [ N∏i=1

qi

] N (q21 , . . . q

2N ) , (29)

with βN = (−1)N (N−1)

2 /[1! · · · (2N − 1)!√N !]. At this stage we notice that the propagatorgoes like

G0(ε, T |x, ε) � β2N T

− 12 N (2N+1)

[N∏i=1

ε2i

] 2

N

(ε21 , . . . , ε

2N

)BN ,

BN =(2

π

)N ∫ ∞

0. . .

∫ ∞

0dq1 . . . dqNe

− 12 q

2

[N∏i=1

q2i

] 2

N (q21 , . . . , q2N )

(30)

where the constant BN is derived in Appendix 4.For reflecting boundary conditions the expression for the propagator is the same but with

theSlater determinant given by�(0)q (x) = (1/

√N !) det

1≤i, j≤N[cos(qi x j )]. Using the expansion

formula for the Slater determinant we arrive at

�(0)q (x/

√T ) = 1√

N !∑s∈UN

1

(2s1)! · · · (2sN )! det1≤i, j≤N

[T−s j x2s ji ] det

1≤i, j≤N[q2s ji (−1)s j ] ,

(31)

123


with s = (s1, . . . , sN ). Noticing that the lowest contribution is obtained by of si = i − 1, weobtain

�(0)q

(x√T

)� βN T− N (N−1)

2 N(x21 , . . . , x

2N

) [ N∏i=1

qi

] N (q21 , . . . q

2N ) , (32)

with βN = (−1)N (N−1)

2 /[0! · · · (2(N − 1))!√N !]. At this stage we notice that the propagatorgoes like

G0(ε, T |x, ε) � β2N T

− N (2N−1)2 2

N

(ε21 , . . . , ε

2N

)FN ,

FN = ( 2π

)N ∫∞0 · · · ∫∞

0 dq1 · · · dqNe− 12 q

2 2

N (q21 , · · · , q2N ) (33)

where the constant FN is derived in Appendix 1.

Appendix 3: Meander

For the case of a meander of N vicious walkers (a star with a wall) we need to integrate overthe final position x. Using the Taylor expansion of the Slater determinant as explained in themain text we can write∫

W N

dN xG(1)N (x, T |ε, 0) =

∑n

∫W N

dN x �(1)n (x)�

(1)n (ε)e−E (1)

n T

� (2λ)N23 γN

[N∏i=1

εi

] N (ε21 , . . . ε

2N )∑n

BN (αn1 , . . . , αnN ) N (αn1 , . . . αnN )e−E (1)n T ,

(34)

where we have defined

BN (αn1 , . . . , αnN ) = 1N∏i=1

Ai′(−αni )

∫W N


[Ai(x j − αni )

]. (35)

Similarly, the propagator in the denominator goes like∫W N

dN xG(0)N (x, T |ε, 0) � T− N2

2 γN FN

[N∏i=1

εi

] N (ε21 , . . . ε

2N ) (36)

where FN is a constant with no simple expression.For reflecting boundary conditionswehave instead the following expression for the numerator∫

W N

dN xG(1)N (x, T |ε, 0) =

∑n

∫W N

dN x�(1)n (x)�

(1)n (ε)e−E (1)

n T

= δN

N ! N (x21 , . . . , x2N )∑n

N (βn1 , . . . , βnN )

βn1 · · · βnNe−E (1)

n T (2λ)N (2N−1)

6 (2λ)N/6

∫W N

dN x1∏N

i=1 Ai(−βni )det

1≤i, j≤N[Ai((2λ)1/3x j − βni )]

= δN

N ! N (x21 , . . . , x2N )(2λ)

N (2N−1)6 −N/6

∑n

N (βn1 , . . . , βnN )

βn1 · · · βnNe−E (1)

n T

∫W N

dN z1∏N

i=1 Ai(−βni )det

1≤i, j≤N[Ai(z j − βni )]

123


= δN

N ! N (x21 , . . . , x2N )(2λ)

N (N−1)3

∑n

N (βn1 , . . . , βnN )

βn1 · · · βnNe−E (1)

n T

1∏Ni=1 Ai(−βni )

∫W N

dN z det1≤i, j≤N

[Ai(z j − βni )] (37)

Appendix 4: On the Normalisation Constants

Here we derive the expressions for the two normalisation constants for the case of excursionsfor absorbing and reflecting boundary conditions. For both cases we recall the well-knownresult of Selberg’s integral [13]:

1

N !∫ ∞

0· · ·∫ ∞

0| (x1, . . . , xN )|2c

N∏i=1

xa−1i e−xi dxi =

N−1∏j=0

�(a + jc)�(( j + 1)c)

�(c)

(38)

For the constant BN we do the change of variables q2i /2 = yi so that qidqi = dyi or

dqi = dyi√2yi

, so we can write

BN =(2

π

)N ∫ ∞

0· · ·∫ ∞

0dq1 · · · dqN e− 1

2 q2

[N∏i=1

q2i

] 2

N (q21 , · · · , q2N )

=(2

π

)N

2−N/22N2N (N−1)∫ ∞

0· · ·∫ ∞

0dy1 · · · dyN

N∏i=1

e−yi y−1/2i

[N∏i=1

yi

] 2

N (y1, · · · , yN )

= 212 N (2N+1)

πNN !

N−1∏j=0

�

(3

2+ j

)�( j + 1) = 2

12 N (2N+1)

πN

N−1∏j=0

�(2 + j)�

(3

2+ j

)(39)

Similarly for the case of reflecting boundary conditions we have

FN =(2

π

)N 1

2N/2 2N (N−1)

∫ ∞

0· · ·∫ ∞

0dy1 · · · dyN 2

N (y1, · · · , yN )

N∏i=1

y−1/2i e−yi

= N !2N (2N−1)

2

πN

N−1∏j=0

�

(1

2+ j

)� ( j + 1) = 2

N (2N−1)2

πN

N−1∏j=0

�

(1

2+ j

)� ( j + 2) (40)

Appendix 5: Derivatives of F(x, γ )

Starting from the following expression for F(x, γ ):

F(x, γ ) =√3

x√

πu2/3(x, γ )e−u(x,γ )U

(1

6,4

3, u(x, γ )

), u(x, γ ) = 2γ 3

27x2, (41)

and using the following property

x∂U (a, b, x)

∂x= (a − b + x)U (a, b, x) −U (a − 1, b, x) , (42)

123


we notice that we can write the expression

∂n F(x, γ )

∂xn=

√3

xn+1√

πu2/3(x, γ )e−u(x,γ )

n∑�=0

C (n)� U

(1

6− �,

4

3, u(x, γ )

). (43)

for some set of coefficients {C (n)� } still to be determined. The expression (43) is certainly

correct for n = 1 and n = 2. Let us them assume is holds for any n and performe one morederivative with respect to n:

∂n+1F(x, γ )

∂xn+1 = 1

xn+2√3π

u2/3(x, γ )e−u(x,γ )

n∑�=0

C (n)�

[(−7 − 3n + 6u(x, γ ))U

(1

6− �,

4

3, u(x, γ )

)

−6u(x, γ )U ′(1

6− �,

4

3, u(x, γ )

)]. (44)

But using the property (42), we can write

u(x, γ )U ′(1

6− �,

4

3, u(x, γ )

)=(1

6− � − 4

3+ u

)

U

(1

6− �,

4

3, u(x, γ )

)−U

(1

6− � − 1,

4

3, u(x, γ )

). (45)

Gathering results we find

∂n+1F(x, γ )

∂xn+1 =√3

xn+2√

πu2/3(x, γ )e−u(x,γ )

n∑�=0

C (n)�

[(−n + 2�)U

(1

6− �,

4

3, u(x, γ )

)+ 2U

(1

6− � − 1,

4

3, u(x, γ )

)]. (46)

On the other hand, we want to write this result as (43) for n → n+ 1. This implies to rewritethe sum as:

n∑�=0

C (n)�

[(−n + 2�)U

( 16 − �, 4

3 , u(x, γ ))+ 2U

( 16 − � − 1, 4

3 , u(x, γ ))]

=n∑

�=0C (n)

� (−n + 2�)U( 16 − �, 4

3 , u(x, γ ))+ 2

k∑�=0

C (n)� U

( 16 − � − 1, 4

3 , u(x, γ ))

=n∑

�=0C (n)

� (−n + 2�)U( 16 − �, 4

3 , u(x, γ ))+ 2

n+1∑�=1

C (n)�−1U

( 16 − �, 4

3 , u(x, γ ))

=n+1∑�=0

C (n+1)� U

( 16 − �, 4

3 , u(x, γ ))

. (47)

This results in the following set of recurrence relations for the set of coefficents {C (n)� }:

C (n+1)0 = −nC (n)

0 ,

C (n+1)� = C (n)

� (−n + 2�) + 2C (n)�−1 , � = 1, . . . , n ,

C (n+1)n+1 = 2C (n)

n , (48)

123


with the initial condition C (0)0 = 1. By checking explicitly the value of some of these

coeffcients, and with the help of the Sloane database2, we arrive at the solution:

C (n)� = n!

2n−2�(n − �)!(2� − n)! , � = 0, . . . , n . (49)

A Similar analysis can be perform ny doing derivatives with respect to γ . Starting from (43)one notices that

∂k+n F(x, γ )

∂γ k∂xn=

√3

xn+1√


n∑�=0

k∑s=0

C (n)� D(k)

s (�)U

(1

6− � − s,

4

3, u(x, γ )

), (50)

for some set of coefficients {D(k)s (�)}. Performing one more derivative with respect to γ

allows us to arrive at the following set of recurrence relations for those, viz.

D(k+1)0 (�) = −

(3

2+ k + 3�

)D(k)0 (�) ,

D(k+1)s (�) = −

(3

2+ k + 3� + 3s

)D(k)s (�) − 3D(k)

s−1(�) , s = 1, . . . , k ,

D(k+1)k+1 (�) = −3D(k)

k (�) , (51)

with the initial condition D(0)0 (�) = 1. As we are only interested in the orders k = 0 and

k = 2, we do not need a general solution. This yields

D(0)0 (�) = 1 ,

D(1)0 (�) = −3

2(1 + 2�) , D(1)

1 (�) = −3

D(2)0 (�) = 3

4(1 + 2�)(5 + 6�) , D(2)

1 (�) = 3(7 + 6�) , D(2)2 (�) = 9 . (52)

Appendix 6: Moments

We first recall the following identity∫ ∞

0dssμe−xs = �(μ + 1)

x1+μ. (53)

From here we write

M−(1+μ) =∫ ∞

0dxx−(1+μ)F(x)

= 1

�(μ + 1)

∫ ∞

0dxF(x)

∫ ∞

0dssμe−xs

= 1

�(μ + 1)

∫ ∞

0dssμ

∫ ∞

0dxF(x)e−xs

2 At https://oeis.org

123

https://oeis.org


= 1

�(μ + 1)

∫ ∞

0dssμ F(s)

= 1

�(μ + 1)

∫ ∞

0dssμ+Ee−as2/3 (54)

But ∫ ∞

0dssμ+Ee−as2/3 =

∫ ∞

0ds(s2/3)

32 (μ+E)e−as2/3

= 3

2

∫ ∞

0dyy

32 (μ+E)+1/2e−ay

= 3

2a− 3

2 (μ+E+1)∫ ∞

0dyy

32 (μ+E)+1/2e−y

= 3

2a− 3

2 (μ+E+1)�

(3

2(μ + E + 1)

)(55)

Thus denoting ν = 1 + μ we finally have

M−ν = 3

2a− 3

2 (ν+E)�( 32 (ν + E)

)�(ν)

(56)

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a generalised airy distribution function for the accumulated...

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